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SUMMARY:Kirsten Eisenträger (Penn State University)
DTSTART:20221020T180000Z
DTEND:20221020T190000Z
DTSTAMP:20260423T053131Z
UID:OLS/104
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OLS/104/">A 
 topological approach to undefinability in algebraic extensions of the rati
 onals</a>\nby Kirsten Eisenträger (Penn State University) as part of Onli
 ne logic seminar\n\n\nAbstract\nIn 1970 Matiyasevich proved that Hilbert
 ’s Tenth Problem over the\nintegers is undecidable\, building on work by
  Davis-Putnam-Robinson.\nHilbert’s Tenth Problem over the rationals is s
 till open\, but it could\nbe resolved by giving an existential definition 
 of the integers inside\nthe rationals.\n\nProving whether such a definitio
 n exists is still out of reach. However\,\nwe will show that only “very 
 few” algebraic extensions of the rationals\nhave the property that their
  ring of integers are existentially or\nuniversally definable. Equipping t
 he set of all algebraic extensions of\nthe rationals with a natural topolo
 gy\, we show that only a meager subset\nhas this property.  An important t
 ool is a new normal form theorem for\nexistential definitions in such exte
 nsions.  As a corollary\, we\nconstruct countably many distinct computable
  algebraic extensions whose\nrings of integers are neither existentially n
 or universally definable.\nJoint work with Russell Miller\, Caleb Springer
 \, and Linda Westrick.\n
LOCATION:https://researchseminars.org/talk/OLS/104/
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